Energy and momentum of photons in gravitational field

We study a problem of energy and momentum of photons redshifted in a gravitational field. Based on the Einstein-Maxwell’s equations for electromagnetic waves in curved spacetimes, we derive formulas for the speed, energy and momentum of photons in gravitational fields. The formulas are further specified for the Schwarzschild metric describing a local gravitational field around a massive body. It is shown that energy of photons is conserved in inertial (free-falling) as well as in non-inertial coordinate systems, and no energy is exchanged between photons and the gravitational field. The Planck energy-frequency relation valid in Special Relativity is modified to be applicable also to General Relativity. According to the new Planck relation, the photon energy depends not only on the frequency of photons but also on their speed. In a free-falling system, the photon energy is conserved, because no frequency shift and no change of the photon speed is detected. In non-inertial systems, the photon energy is also conserved, because the frequency shift due to gravity is compensated by the change of the photon speed.